Regularization properties of Mumford–Shah-type functionals with perimeter and norm constraints for linear ill-posed problems

In this paper we consider the simultaneous reconstruction and segmentation of a function f from measurements g = Kf, where K is a linear operator. Assuming that the inversion of K is illposed, regularization methods have to be used for the inversion process in case of inexact data. We propose using a Mumford–Shah-type functional for the stabilization of the inversion. Restricting our analysis to the recovery of piecewise constant functions, we investigate the existence of minimizers, their stability, and the regularization properties of our approach. Finally, we present a numerical example from single photon emission computed tomography (SPECT).FWF, T 529-N18, Mumford-Shah models for tomography I

An alternating iterative minimisation algorithm for the double-regularised total least square functional

The total least squares (TLS) method is a successful approach for linear problems if both the right-hand side and the operator are contaminated by some noise. For ill-posed problems, a regularisation strategy has to be considered to stabilise the computed solution. Recently a double regularised TLS method was proposed within an infinite dimensional setup and it reconstructs both function and operator, reflected on the bilinear forms Our main focuses are on the design and the implementation of an algorithm with particular emphasis on alternating minimisation strategy, for solving not only the double regularised TLS problem, but a vast class of optimisation problems: on the minimisation of a bilinear functional of two variables.Peer reviewe